File:Archimedes1.svg

Summary
Archimedes used a diagram like this to determine the ratio of a circle's circumference to its diameter by inscribing successively complex polygons that shared either all vertices or all segment midpoints with a circle of diameter 1. Once he arrived at one polygon sharing 96 vertices and another polygon sharing 96 segment midpoints with the circle, he then averaged the perimeter of the two 96-gons and arrived at 3.14, the number we know as π.